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Properties

Label 1323.2.h.e.226.2
Level $1323$
Weight $2$
Character 1323.226
Analytic conductor $10.564$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1323,2,Mod(226,1323)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1323.226"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,2,0,6,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 1323.226
Dual form 1323.2.h.e.802.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.239123 q^{2} -1.94282 q^{4} +(-0.590972 - 1.02359i) q^{5} -0.942820 q^{8} +(-0.141315 - 0.244765i) q^{10} +(-1.85185 + 3.20750i) q^{11} +(0.500000 - 0.866025i) q^{13} +3.66019 q^{16} +(3.47141 + 6.01266i) q^{17} +(0.971410 - 1.68253i) q^{19} +(1.14815 + 1.98866i) q^{20} +(-0.442820 + 0.766987i) q^{22} +(-2.80150 - 4.85235i) q^{23} +(1.80150 - 3.12030i) q^{25} +(0.119562 - 0.207087i) q^{26} +(0.119562 + 0.207087i) q^{29} -1.66019 q^{31} +2.76088 q^{32} +(0.830095 + 1.43777i) q^{34} +(4.77292 - 8.26693i) q^{37} +(0.232287 - 0.402332i) q^{38} +(0.557180 + 0.965064i) q^{40} +(5.09097 - 8.81782i) q^{41} +(-1.11273 - 1.92730i) q^{43} +(3.59781 - 6.23159i) q^{44} +(-0.669905 - 1.16031i) q^{46} +5.82846 q^{47} +(0.430782 - 0.746136i) q^{50} +(-0.971410 + 1.68253i) q^{52} +(-5.80150 - 10.0485i) q^{53} +4.37756 q^{55} +(0.0285900 + 0.0495193i) q^{58} +2.60301 q^{59} +7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} -1.18194 q^{65} +3.50808 q^{67} +(-6.74433 - 11.6815i) q^{68} -8.60301 q^{71} +(7.57442 + 13.1193i) q^{73} +(1.14132 - 1.97682i) q^{74} +(-1.88727 + 3.26886i) q^{76} +7.37756 q^{79} +(-2.16307 - 3.74654i) q^{80} +(1.21737 - 2.10855i) q^{82} +(3.47141 + 6.01266i) q^{83} +(4.10301 - 7.10662i) q^{85} +(-0.266078 - 0.460861i) q^{86} +(1.74596 - 3.02409i) q^{88} +(-1.37360 + 2.37915i) q^{89} +(5.44282 + 9.42724i) q^{92} +1.39372 q^{94} -2.29630 q^{95} +(3.58414 + 6.20790i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + 12 q^{8} - 2 q^{11} + 3 q^{13} + 6 q^{16} + 12 q^{17} - 3 q^{19} + 16 q^{20} + 15 q^{22} - 6 q^{25} + q^{26} + q^{29} + 6 q^{31} + 16 q^{32} - 3 q^{34} + 3 q^{37} - 8 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.239123 0.169086 0.0845428 − 0.996420i \(-0.473057\pi\)
0.0845428 + 0.996420i \(0.473057\pi\)
\(3\) 0 0
\(4\) −1.94282 −0.971410
\(5\) −0.590972 − 1.02359i −0.264291 − 0.457765i 0.703087 − 0.711104i \(-0.251804\pi\)
−0.967378 + 0.253339i \(0.918471\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.942820 −0.333337
\(9\) 0 0
\(10\) −0.141315 − 0.244765i −0.0446878 − 0.0774015i
\(11\) −1.85185 + 3.20750i −0.558353 + 0.967096i 0.439281 + 0.898350i \(0.355233\pi\)
−0.997634 + 0.0687465i \(0.978100\pi\)
\(12\) 0 0
\(13\) 0.500000 − 0.866025i 0.138675 − 0.240192i −0.788320 − 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.66019 0.915047
\(17\) 3.47141 + 6.01266i 0.841941 + 1.45828i 0.888252 + 0.459357i \(0.151920\pi\)
−0.0463112 + 0.998927i \(0.514747\pi\)
\(18\) 0 0
\(19\) 0.971410 − 1.68253i 0.222857 − 0.385999i −0.732818 − 0.680425i \(-0.761795\pi\)
0.955674 + 0.294426i \(0.0951285\pi\)
\(20\) 1.14815 + 1.98866i 0.256735 + 0.444677i
\(21\) 0 0
\(22\) −0.442820 + 0.766987i −0.0944096 + 0.163522i
\(23\) −2.80150 − 4.85235i −0.584154 − 1.01178i −0.994980 − 0.100071i \(-0.968093\pi\)
0.410826 − 0.911714i \(-0.365240\pi\)
\(24\) 0 0
\(25\) 1.80150 − 3.12030i 0.360301 − 0.624060i
\(26\) 0.119562 − 0.207087i 0.0234480 − 0.0406131i
\(27\) 0 0
\(28\) 0 0
\(29\) 0.119562 + 0.207087i 0.0222020 + 0.0384551i 0.876913 − 0.480649i \(-0.159599\pi\)
−0.854711 + 0.519104i \(0.826266\pi\)
\(30\) 0 0
\(31\) −1.66019 −0.298179 −0.149089 − 0.988824i \(-0.547634\pi\)
−0.149089 + 0.988824i \(0.547634\pi\)
\(32\) 2.76088 0.488059
\(33\) 0 0
\(34\) 0.830095 + 1.43777i 0.142360 + 0.246575i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.77292 − 8.26693i 0.784662 − 1.35908i −0.144538 − 0.989499i \(-0.546170\pi\)
0.929201 − 0.369576i \(-0.120497\pi\)
\(38\) 0.232287 − 0.402332i 0.0376819 − 0.0652669i
\(39\) 0 0
\(40\) 0.557180 + 0.965064i 0.0880979 + 0.152590i
\(41\) 5.09097 − 8.81782i 0.795076 − 1.37711i −0.127715 − 0.991811i \(-0.540764\pi\)
0.922791 − 0.385301i \(-0.125903\pi\)
\(42\) 0 0
\(43\) −1.11273 − 1.92730i −0.169689 − 0.293910i 0.768622 − 0.639704i \(-0.220943\pi\)
−0.938311 + 0.345794i \(0.887610\pi\)
\(44\) 3.59781 − 6.23159i 0.542390 − 0.939447i
\(45\) 0 0
\(46\) −0.669905 − 1.16031i −0.0987721 − 0.171078i
\(47\) 5.82846 0.850168 0.425084 − 0.905154i \(-0.360245\pi\)
0.425084 + 0.905154i \(0.360245\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.430782 − 0.746136i 0.0609217 − 0.105520i
\(51\) 0 0
\(52\) −0.971410 + 1.68253i −0.134710 + 0.233325i
\(53\) −5.80150 − 10.0485i −0.796898 − 1.38027i −0.921627 − 0.388077i \(-0.873139\pi\)
0.124729 − 0.992191i \(-0.460194\pi\)
\(54\) 0 0
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) 0 0
\(58\) 0.0285900 + 0.0495193i 0.00375405 + 0.00650220i
\(59\) 2.60301 0.338883 0.169442 − 0.985540i \(-0.445804\pi\)
0.169442 + 0.985540i \(0.445804\pi\)
\(60\) 0 0
\(61\) 7.60301 0.973466 0.486733 − 0.873551i \(-0.338189\pi\)
0.486733 + 0.873551i \(0.338189\pi\)
\(62\) −0.396990 −0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) −1.18194 −0.146602
\(66\) 0 0
\(67\) 3.50808 0.428580 0.214290 − 0.976770i \(-0.431256\pi\)
0.214290 + 0.976770i \(0.431256\pi\)
\(68\) −6.74433 − 11.6815i −0.817870 − 1.41659i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.60301 −1.02099 −0.510495 − 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(72\) 0 0
\(73\) 7.57442 + 13.1193i 0.886519 + 1.53550i 0.843963 + 0.536402i \(0.180217\pi\)
0.0425559 + 0.999094i \(0.486450\pi\)
\(74\) 1.14132 − 1.97682i 0.132675 − 0.229800i
\(75\) 0 0
\(76\) −1.88727 + 3.26886i −0.216485 + 0.374963i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.37756 0.830040 0.415020 − 0.909812i \(-0.363775\pi\)
0.415020 + 0.909812i \(0.363775\pi\)
\(80\) −2.16307 − 3.74654i −0.241838 − 0.418876i
\(81\) 0 0
\(82\) 1.21737 − 2.10855i 0.134436 − 0.232850i
\(83\) 3.47141 + 6.01266i 0.381037 + 0.659975i 0.991211 − 0.132292i \(-0.0422338\pi\)
−0.610174 + 0.792267i \(0.708900\pi\)
\(84\) 0 0
\(85\) 4.10301 − 7.10662i 0.445034 − 0.770821i
\(86\) −0.266078 − 0.460861i −0.0286920 − 0.0496960i
\(87\) 0 0
\(88\) 1.74596 − 3.02409i 0.186120 − 0.322369i
\(89\) −1.37360 + 2.37915i −0.145602 + 0.252189i −0.929597 − 0.368577i \(-0.879845\pi\)
0.783996 + 0.620766i \(0.213178\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.44282 + 9.42724i 0.567453 + 0.982858i
\(93\) 0 0
\(94\) 1.39372 0.143751
\(95\) −2.29630 −0.235596
\(96\) 0 0
\(97\) 3.58414 + 6.20790i 0.363914 + 0.630317i 0.988601 − 0.150558i \(-0.0481069\pi\)
−0.624687 + 0.780875i \(0.714774\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1323.2.h.e.226.2 6
3.2 odd 2 441.2.h.b.373.2 6
7.2 even 3 1323.2.f.c.442.2 6
7.3 odd 6 1323.2.g.c.361.2 6
7.4 even 3 1323.2.g.b.361.2 6
7.5 odd 6 189.2.f.a.64.2 6
7.6 odd 2 1323.2.h.d.226.2 6
9.2 odd 6 441.2.g.d.79.2 6
9.7 even 3 1323.2.g.b.667.2 6
21.2 odd 6 441.2.f.d.148.2 6
21.5 even 6 63.2.f.b.22.2 ✓ 6
21.11 odd 6 441.2.g.d.67.2 6
21.17 even 6 441.2.g.e.67.2 6
21.20 even 2 441.2.h.c.373.2 6
28.19 even 6 3024.2.r.g.1009.3 6
63.2 odd 6 441.2.f.d.295.2 6
63.5 even 6 567.2.a.d.1.2 3
63.11 odd 6 441.2.h.b.214.2 6
63.16 even 3 1323.2.f.c.883.2 6
63.20 even 6 441.2.g.e.79.2 6
63.23 odd 6 3969.2.a.m.1.2 3
63.25 even 3 inner 1323.2.h.e.802.2 6
63.34 odd 6 1323.2.g.c.667.2 6
63.38 even 6 441.2.h.c.214.2 6
63.40 odd 6 567.2.a.g.1.2 3
63.47 even 6 63.2.f.b.43.2 yes 6
63.52 odd 6 1323.2.h.d.802.2 6
63.58 even 3 3969.2.a.p.1.2 3
63.61 odd 6 189.2.f.a.127.2 6
84.47 odd 6 1008.2.r.k.337.2 6
252.47 odd 6 1008.2.r.k.673.2 6
252.103 even 6 9072.2.a.cd.1.1 3
252.131 odd 6 9072.2.a.bq.1.3 3
252.187 even 6 3024.2.r.g.2017.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 21.5 even 6
63.2.f.b.43.2 yes 6 63.47 even 6
189.2.f.a.64.2 6 7.5 odd 6
189.2.f.a.127.2 6 63.61 odd 6
441.2.f.d.148.2 6 21.2 odd 6
441.2.f.d.295.2 6 63.2 odd 6
441.2.g.d.67.2 6 21.11 odd 6
441.2.g.d.79.2 6 9.2 odd 6
441.2.g.e.67.2 6 21.17 even 6
441.2.g.e.79.2 6 63.20 even 6
441.2.h.b.214.2 6 63.11 odd 6
441.2.h.b.373.2 6 3.2 odd 2
441.2.h.c.214.2 6 63.38 even 6
441.2.h.c.373.2 6 21.20 even 2
567.2.a.d.1.2 3 63.5 even 6
567.2.a.g.1.2 3 63.40 odd 6
1008.2.r.k.337.2 6 84.47 odd 6
1008.2.r.k.673.2 6 252.47 odd 6
1323.2.f.c.442.2 6 7.2 even 3
1323.2.f.c.883.2 6 63.16 even 3
1323.2.g.b.361.2 6 7.4 even 3
1323.2.g.b.667.2 6 9.7 even 3
1323.2.g.c.361.2 6 7.3 odd 6
1323.2.g.c.667.2 6 63.34 odd 6
1323.2.h.d.226.2 6 7.6 odd 2
1323.2.h.d.802.2 6 63.52 odd 6
1323.2.h.e.226.2 6 1.1 even 1 trivial
1323.2.h.e.802.2 6 63.25 even 3 inner
3024.2.r.g.1009.3 6 28.19 even 6
3024.2.r.g.2017.3 6 252.187 even 6
3969.2.a.m.1.2 3 63.23 odd 6
3969.2.a.p.1.2 3 63.58 even 3
9072.2.a.bq.1.3 3 252.131 odd 6
9072.2.a.cd.1.1 3 252.103 even 6